Analyzing Shifts in \(A\) and \(\alpha\)#
Shifts in \(A\) and their Effect on Output#
First, let us plot a 3D surface of the Cobb-Douglas production function. Output, \(Y\) , will go on the vertical (or \(z\) ) axis. Capital and labor will go on the \(y\) and \(x\) axes, resp. The plot below plots the Cobb-Douglas function with \(A=2\), also showing \(A=1\) for reference.
[Following image is a 3D plot of Output increasing with Capital and Labor]
Supply or total factor productivity shocks could cause \(A\) to change. These occur if there is a change in total output for a given level of capital and labor. Examples of these include financial crises, technology shocks, natural environment/distasters and energy prices.
[Following image is an interactive 3D plot of Output increasing with Capital and Labor as A changes]
Favorable shocks rotate the production function upward through an increase in A. Thus, each unit of input from capital and labor now simulataneously produce more output. What does this mean for the rental rate of capital and the real wage? Recall the functions for both of them:
Both MPK and MPL will increase by a factor of \(A\). Thus, it would be more expensive to hire an additional unit of labor or rent an additional unit of capital. As they are both more productive than previously, they are both more valuable to a business and thus will cost more. Negative shocks do the opposite. They rotate the production function downward through a decrease in \(A\). Each unit of input is now less productive, meaning that both the rental rate of capital and the real wage are lower.
Shifts in \(\alpha\) and their Effect on Output#
We will now plot what happens to the Cobb-Douglas function as we vary \(\alpha\), while holding all other variables constant. The plot below shows \(\alpha = 0.8\) (the purple-yellow surface) with \(\alpha=0.5\) for reference (the blue-yellow surface). Try and hypothesize what this will do to our production function.
[Following image is a 3D plot of Output increasing with Capital and Labor, with shifts as alpha changes]